A "change of basis" is an action performed in linear algebra, whereby a change in fundamental structure yields an entirely new viewpoint. This blog began as a record of a pedagogical change of basis for me, and continues as an ongoing account of my thoughts as I design and direct courses in mathematics at the University of North Carolina, Asheville.

Wednesday, September 13, 2006

Today saw us going over matrix inverses again, more fully than we'd done on Friday. I felt that Friday we'd not yet had a chance to really get a good grip on the geometric meaning of an inverse, so we retraced our steps into that territory before learning how to run a Markov process in reverse.

Although I think most of the folks in the class got the hang of it by the end of the class, there were a few who remained confused. (One student came by after class, frustrated as hell at not picking up on a lot of what we'd done. We spent about fifteen minutes working through the applications we'd considered, one-on-one. By the end I think it was clear that the student understood more of what was going on than the student had thought at first.)

Not knowing what in the wide world of wackiness is going on is truly frustrating.

And there's no doubt about it: frustration sucks.

Frustration works its ways differently on different people. Some cry, some scream, some just lose all capacity for rational thought.

The most frustrating incident of my academic career came in my third year of graduate school, at the end of the Fall 2000 semester at Vanderbilt.

That'd been a tough semester. If memory serves, I had a somewhat traditional algebraic topology course, a couple of seminar classes, including one on combinatorial group theory and another on small cancellation theory, and I was auditing a class on advanced group theory. Meanwhile I was teaching a pretty cool Calc I class, helping to put together our department's undergraduate seminar in mathematics and compose our in-house precalc review text, and conducting research that would later become my first published article.

By early December, I was pretty much gone.

The seminar on small cancellation theory was an exciting one. The instructor, one Professor Peanut, was (and is) one of the world's foremost experts on combinatorial group theory, universally respected in his area, and well-known outside of mathematics for his theories on the use of mathematics in deciphering biblical prophecy. Big, big man in the mathematical community. A very kind gentleman, he terrified me nonetheless with his stature and renown.

He was a man of deep conviction and numerous idiosyncrasies. An orthodox Jew, he davened constantly while reading his Torah, something he did during just about every break available to him. He played with his beard absentmindedly, stroking its great gray length with thick stubby fingers. He drank Diet Coke as though he owned stock in it. If he were not the one speaking but only participating as an audience member, he would sit in the front row (an open can of Diet Coke in front of him), his attention fixed on the speaker for all of about ten minutes before sleep overtook him and he dozed off, his chin sinking through his beard to rest upon his chest. He would sleep for five or six minutes at a time before waking and looking around as though to make sure no one had caught him in his nap. Five minutes later the cycle began anew.

Strange travel restrictions forced him to spend with us only half of the semester he'd promised our department, so it was not until mid-October that he came to the United States to begin the seminar course we'd scheduled for the fall. Once underway, we (the five of us registered for the course and a few faculty hangers-on who wanted to sit in) decided the best way to make up for lost time would be to meet once a week for twice as long as we'd originally meant to meet. So began our three-hour Thursdays.

The class was fascinating. Peanut would lecture to us extemporaneously on cutting-edge research, much of which he'd developed only weeks or even days before. Unlike his long-time colleague and co-author, Professor Cashew, who for the past year had been a full-time faculty member at Vanderbilt, Peanut was not a slave to detail and would almost invariably omit precise computations and rigorous proofs, opting instead to give outlines and general descriptions of his intricate arguments. Though he painted in broad strokes, the pictures he produced seemed clear to us, and only occasionally would we have need to stop him in his exposition. Much of the beauty of math lies in the creativity that goes into the construction of new techniques, and Peanut's were among the most delicate and delicious I've ever seen. Though his exposition was too sparse to allow me to get at the innermost workings of his mathematical thought, I felt I had a good understanding of the subject.

I realize now how difficult it is to learn to paint without actually getting the chance to hold the brush yourself.

We spent nine or ten evenings in these seminars, and in that time I filled perhaps four dozen pages with notes on Peanut's techniques. Supplementing this were about thirty pages of poorly mimeographed notes Peanut himself had written (by hand, in very nice and readable Russian script) in preparation for writing one of his papers. We had a good written record of the course, but given its lack of precision (like his speech, my notes are filled with phrases like "quite big," "big enough," "very small," and so forth), to do much math with what we had would be like to building a warp drive from pencils and chewing gum after watching an episode of Deep Space Nine.

I must take a moment to describe one of the course's faculty fans. Professor Pistachio was (and is) a brilliant man with an acerbic wit and a certain impishness that came out most noticeably when torturing undergraduates. On his website he keeps a page which admits that his students often complain he is too sparing with positive feedback. To counter this, he includes on the page a MIDI file of soothing music, after several seconds of which he intones carefully and lovingly, "good job. Good job. You like the word 'great' better? Great job..."

"Sarcastic" doesn't cover it. (In his defense, I must say that I consider Pistachio one of my favorite professors in graduate school, an excellent expositor of mathematics, and one of the best writers of mathematics I've ever known: it's hard to write clear technical mathematics articles, and he does this exceedingly well.)

At the end of the last class, Peanut dismissed us. "That is all," he said, and with a kindly smile he set down the chalk. If only it were so! From his seat to my left, Professor Pistachio said, in Russian, "what about their exam?"

We sank. We'd expected to not be tested; it was an unwritten understanding. After all, this was a seminar course, and those who participated did so because they wanted to learn the material. We weren't undergrads anymore, we didn't have to be prodded and cajoled by some penny-ante system of praise and punishment. We didn't need an exam!!!

More to the point, we weren't ready for an exam. What would he, could he ask of us? How could we be tested on details we'd never learned?

"Ah yes," Peanut replied softly. "There is the matter of this exam. I think perhaps it should be...oral examinations. You will each schedule a time to meet me individually, yes?"

We left the room dejectedly. I trudged back to our basement offices with my colleagues Damon, Maurice and Nathaniel. We discussed our plans. "We've got a week," we agreed, "before he leaves the country. We can study. We have notes."

But we were all busy. We had finals to write and proctor and grade, we had research, we had other classes. Our time was precious, and we could spare little of it to digest the abstruse methods we'd been given in big, unchewable chunks over the preceding weeks.

"I'm going to do it tomorrow," I told my friends the next day.

"Daring," they agreed.

"I just want to get it over with," I said. I sent Peanut an e-mail and made an appointment at 4:00 the following afternoon.

I spent the next 24 hours cramming like never before. I read over my notes again, forwards, backwards, sideways, diagonally. Like Peanut with his biblical gematriyot, I sought meaning in every word.

4:00 the next day came way too quickly. I knocked timidly on the open door of the office Peanut shared with a few other visiting faculty members. There was no answer. I knocked again, and again there was no response. I padded into the office softly and peered around the cubicle divider which separated Peanut's desk from the one nearest to the door. At his desk he sat, squinting at the screen of his laptop. "Professor Peanut?" I asked. He looked up and smiled and gestured for me to sit in the chair across from him. I sat, and the test began.

"First, can you tell me how to describe the automorphisms of the...icosahedron?"

What in the HELL?!!?!!? This question had NOTHING to do with ANYTHING we'd talked about in the last several weeks. NOTHING. Nada. Zippo. My mind spun.

"You have had a basic course in group theory?" Peanut asked, his tone belying at least a hint of annoyance.

"Yes," I insisted. "But..." He scribbled a sketch and a hint or two on a piece of paper and thrust it across the desk towards me. I stared at it blankly. "Um...let me think..." And he did. He folded his fingers together and sat back in silence as I struggled to make connections between what I knew of icosahedra and what I'd learned in Peanut's course. There wasn't much to go on, but I managed to retrieve a few tidbits concerning fixed point sets and fundamental domains, and with these I pieced together a good bit of BS.

I looked up. Peanut, true to form, was asleep.

"Professor Peanut?" I asked gently. "Professor Peanut?" He slept on. I leaned back and sighed, and the hot light of the setting sun cut into the room through the slats of the office's Venetian blinds. I squinted and looked outside at the walls of the physics building next door. I wished myself far away.

I tried again to rouse Peanut, and at last succeeded. I showed him my notes, and he nodded and made a few ambiguous and uninterpretable noises. Was I right on or full of crap? I had no idea. As he let me get back to work, I looked wonderingly at the notes I'd produced. I'd been in Peanut's office for almost an hour and had done almost nothing. Quite frankly, I felt like shit.

Peanut let me struggle for another quarter-hour, and at the end he let me off the hook. "Okay," he said. "That is enough." I'd made a half-assed attempt at solving one easy question, and in producing that one solution I'd demonstrated minimal knowledge of a few minor techniques we'd covered in the class. The worst was yet to come.

"Now I must assign a grade," said Peanut. "I'm afraid I do not understand...what does this mean, an 'A'? What does a 'B' mean?" I gave him a briefing on the American grading system, perhaps curving each category a bit too generously. Peanut seemed satisfied. His next question came sock in the gut.

"What grade do you deserve?" he asked.

After the worst academic performance of my life, I'd been asked to assess its merit.

The lowest grade I got in grad school, I gave myself.

I left Peanut's office numbly and dumbly. I went to my office and let the others know how it had gone, and then I left. I spent the rest of the evening in a stupor, eating flavorless food and having mirthless conversations.

Yeah, frustration sucks.

I hope that if you're ever frustrated with this course, like our friend who spent some time with me this afternoon, you'll be willing to share that frustration with me. It feels better when it's spread around, and once it's out there, we can do something about it.

Okay, that's enough for tonight. I'm sure you've all enjoyed this bedtime story. Feel free to post away and share your own moments of frustration! Meanwhile, let's hit that reading for the weekend, and get ready for some hardcore Tinkertoy action on Monday!

6 comments:

Ican Relate
said...

Shazaam...this is the second grad school story in the past few days I've heard. I love hearing about experiences, but now, I'm not really sure if I want to go to graduate school.

As for frustration, that's happened to me several times in our linear algebra class (not directed at anyone, save myself). Reading over the textbook and preparing for class, I always feel like "Okay, I know what I'm doing, I'm ready. Quiz today?, no problem, I can handle it." Then class happens. It's frustrating when people seem to understand with no problem and I'm left thinking "hunh?" I always have to remind myself with my 4th grade teacher said. "If you have a question, usually there is someone else, if not several others, with that very same question." My team is awesome helping me out, but sometimes it may be good to voice a question to the class as a whole, allowing you, Patrick, to get an assessment of the confusion (which is often the seed of frustration); this also may cut down on the chattering banter-murmer which hums throughout class and sometimes gets distracting. I don't want to take away from the individual teams, but the class itself it one big team when you think about it.

Your suggestion, Ican, is a good one, one that strikes a compromise between the team-oriented approach we've adopted and the more traditional classroom.

Another compromise I've meant to make more use of and have not, as yet, is allowing individual students a few moments to singly derive a solution to a given problem. This simple activity, again more common in a traditional classroom than in ours, gives those who might work at a slightly slower pace (or work better alone, without the "chattering banter" of others to distract them!) a chance to come up with their own solutions first. We'll start out class on Monday with such an exercise.

Good...good...food for thought! I'll have to keep your suggestion in mind.

Okay, I'll admit it. I've taken linear algebra before, didn't do so well, so I'm back. The first time around, the professor taught in the traditional way of, every class period, he'd write on the board and talk. Matrices are handy, but after copying a few pages of matrix notes, the newness of the relationship wears off. Everything in your notes becomes hazy, the professor's notes on the board become muddled, and you find yourself anxiously waiting for class to end.

Now, revisiting linear algebra, I'm always shocked when I look down at my watch and see that there's only 5 minutes of class left. True, my notes are not at all traditional, but I comprehend and retain so much more this time around. Perhaps a tiny part is exposure to the material a second time. However, I believe the reason I am understanding linear algebra now is the manner of how the material is presented, and the fact that I and my team are interacting with the material rather than digesting it from examples written on the board. When we have quizzes, I don't feel that I am simply spitting up information I've copied down, but rather, that I'm applying linear algebra to a particular situation.

In short, I like the way the class is being taught. Yes, while in class I do ask my team when I need something clarified rather than asking Patrick, but with the Math Lab and the seemingly endless office hours Patrick keeps, my questions get answered.

I understand that hearing the questions of others could by very valuable; it usually is. But if it's a really major question, usually the team will ask Patrick and he clarifies for the class. I would be okay with people asking more questions in class, but that may affect the team dynamic on which the class is designed. I prefer this class much more that to how I was first introduced to linear algebra.

And, from now on, I'll try and whisper when I need to ask my team to clarify something.

Question for anonymous and aaron: why do you find it shocking when such a small amount of time remains? Is it that time flies by when you're having fun? (he asked hopefully...)

I wanted to make another comment to ican's comment, a very general one regarding graduate school: please know that most folks who go so far as grad school (myself included) consider graduate school the best time of their lives. Those four years were fantastic, and it was there that I made some of the best friends I'll ever have, and made some of the most meaningful observations about myself and how I think and act. It was a wonderful time of my life, and incidents like the one I describe in "Story Time" are wholly exceptional!

your lectures are interesting. almost everything you say is funny. the feeling that you are (somewhat) close to our age group, and usually understand what we're going through, and make an effort to treat us the way we want to be treated help in making class time go a bit more quickly than the normal. usually, professors stand by the board and talk non-stop or work problem after problem after problem, and it is so monotone (<< can't spell), it KILLS ME!! my attention span is probably about 0.2micro-seconds, and your "hyper-ness" suits my jumpy brain perfectly.